A sequence of translation surfaces $(X_n,\omega_n)$ is said to "go to infinity" if it leaves every compact set in the space of translation surfaces as $n$ goes to infinity. I know that this is equivalent to saying that the systole (i.e. the length of the shortest saddle connection) of $(X_n,\omega_n)$ goes to zero. But I'm wondering if the converse condition on the length is true. In particular is it true that the condition of $(X_n,\omega_n)$ going to infinity implies that there exists a $[\gamma]$ in the first group of relative homology of the surface such that the length of $[\gamma]$ with respect of $\omega_n$ goes to infinity?

I am not sure about your definitions, but I think that the answer has to be "no". Consider the regular decagon, with opposite sides identified. Choose a single pair of sides, and collapse those. Then the sequence and its limit $(X, \omega)$ lie in different strata. But $(X, \omega)$ is a well-behaved translation surface, so any class $[\gamma]$ has a finite limit (perhaps of zero length).